2,040 research outputs found
Probability Theory Compatible with the New Conception of Modern Thermodynamics. Economics and Crisis of Debts
We show that G\"odel's negative results concerning arithmetic, which date
back to the 1930s, and the ancient "sand pile" paradox (known also as "sorites
paradox") pose the questions of the use of fuzzy sets and of the effect of a
measuring device on the experiment. The consideration of these facts led, in
thermodynamics, to a new one-parameter family of ideal gases. In turn, this
leads to a new approach to probability theory (including the new notion of
independent events). As applied to economics, this gives the correction, based
on Friedman's rule, to Irving Fisher's "Main Law of Economics" and enables us
to consider the theory of debt crisis.Comment: 48p., 14 figs., 82 refs.; more precise mathematical explanations are
added. arXiv admin note: significant text overlap with arXiv:1111.610
q-Legendre Transformation: Partition Functions and Quantization of the Boltzmann Constant
In this paper we construct a q-analogue of the Legendre transformation, where
q is a matrix of formal variables defining the phase space braidings between
the coordinates and momenta (the extensive and intensive thermodynamic
observables). Our approach is based on an analogy between the semiclassical
wave functions in quantum mechanics and the quasithermodynamic partition
functions in statistical physics. The basic idea is to go from the
q-Hamilton-Jacobi equation in mechanics to the q-Legendre transformation in
thermodynamics. It is shown, that this requires a non-commutative analogue of
the Planck-Boltzmann constants (hbar and k_B) to be introduced back into the
classical formulae. Being applied to statistical physics, this naturally leads
to an idea to go further and to replace the Boltzmann constant with an infinite
collection of generators of the so-called epoch\'e (bracketing) algebra. The
latter is an infinite dimensional noncommutative algebra recently introduced in
our previous work, which can be perceived as an infinite sequence of
"deformations of deformations" of the Weyl algebra. The generators mentioned
are naturally indexed by planar binary leaf-labelled trees in such a way, that
the trees with a single leaf correspond to the observables of the limiting
thermodynamic system
Techniques for the Synthesis of Reversible Toffoli Networks
This paper presents novel techniques for the synthesis of reversible networks
of Toffoli gates, as well as improvements to previous methods. Gate count and
technology oriented cost metrics are used. Our synthesis techniques are
independent of the cost metrics. Two new iterative synthesis procedure
employing Reed-Muller spectra are introduced and shown to complement earlier
synthesis approaches. The template simplification suggested in earlier work is
enhanced through introduction of a faster and more efficient template
application algorithm, updated (shorter) classification of the templates, and
presentation of the new templates of sizes 7 and 9. A novel ``resynthesis''
approach is introduced wherein a sequence of gates is chosen from a network,
and the reversible specification it realizes is resynthesized as an independent
problem in hopes of reducing the network cost. Empirical results are presented
to show that the methods are effective both in terms of the realization of all
3x3 reversible functions and larger reversible benchmark specifications.Comment: 20 pages, 5 figure
Mathematical Conception of "Phenomenological" Equilibrium Thermodynamics
In the paper, the principal aspects of the mathematical theory of equilibrium
thermodynamics are distinguished. It is proved that the points of degeneration
of a Bose gas of fractal dimension in the momentum space coincide with critical
points or real gases, whereas the jumps of critical indices and the Maxwell
rule are related to the tunnel generalization of thermodynamics. Semiclassical
methods are considered for the tunnel generalization of thermodynamics and also
for the second and ultrasecond quantization (operators of creation and
annihilation of pairs). To every pure gas there corresponds a new critical
point of the limit negative pressure below which the liquid passes to a
dispersed state (a foam). Relations for critical points of a homogeneous
mixture of pure gases are given in dependence on the concentration of gases.Comment: 37 pages, 9 figure, more precise explanations, more references. arXiv
admin note: substantial text overlap with arXiv:1202.525
Semiclassical Estimates of Electromagnetic Casimir Self-Energies of Spherical and Cylindrical Metallic Shells
The leading semiclassical estimates of the electromagnetic Casimir stresses
on a spherical and a cylindrical metallic shell are within 1% of the field
theoretical values. The electromagnetic Casimir energy for both geometries is
given by two decoupled massless scalars that satisfy conformally covariant
boundary conditions. Surface contributions vanish for smooth metallic
boundaries and the finite electromagnetic Casimir energy in leading
semiclassical approximation is due to quadratic fluctuations about periodic
rays in the interior of the cavity only. Semiclassically the non-vanishing
Casimir energy of a metallic cylindrical shell is almost entirely due to
Fresnel diffraction.Comment: 12 pages, 2 figure
First-Matsubara-frequency rule in a Fermi liquid. Part I: Fermionic self-energy
We analyze in detail the fermionic self-energy \Sigma(\omega, T) in a Fermi
liquid (FL) at finite temperature T and frequency \omega. We consider both
canonical FLs -- systems in spatial dimension D >2, where the leading term in
the fermionic self-energy is analytic [the retarded Im\Sigma^R(\omega,T) =
C(\omega^2 +\pi^2 T^2)], and non-canonical FLs in 1<D <2, where the leading
term in Im\Sigma^R(\omega,T) scales as T^D or \omega^D. We relate the \omega^2
+ \pi^2 T^2 form to a special property of the self-energy -"the
first-Matsubara-frequency rule", which stipulates that \Sigma^R(i\pi T,T) in a
canonical FL contains an O(T) but no T^2 term. We show that in any D >1 the
next term after O(T) in \Sigma^R(i\pi T,T) is of order T^D (T^3\ln T in D=3).
This T^D term comes from only forward- and backward scattering, and is
expressed in terms of fully renormalized amplitudes for these processes. The
overall prefactor of the T^D term vanishes in the "local approximation", when
the interaction can be approximated by its value for the initial and final
fermionic states right on the Fermi surface. The local approximation is
justified near a Pomeranchuk instability, even if the vertex corrections are
non-negligible. We show that the strength of the first-Matsubara-frequency rule
is amplified in the local approximation, where it states that not only the T^D
term vanishes but also that \Sigma^R(i\pi T,T) does not contain any terms
beyond O(T). This rule imposes two constraints on the scaling form of the
self-energy: upon replacing \omega by i\pi T, Im\Sigma^R(\omega,T) must vanish
and Re\Sigma^R (\omega, T) must reduce to O(T). These two constraints should be
taken into consideration in extracting scaling forms of \Sigma^R(\omega,T) from
experimental and numerical data.Comment: 22 pages, 3 figure
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